A134174 -id:A134174 - cp's OEIS frontend

A163353 G.f.: A(x,y) = Sum_{n>=0,m>=0} (2^m-1)^nx^n log(1+y)^m/m!.

1, 1, 0, 1, 1, 0, 1, 4, 4, 1, 0, 1, 13, 44, 67, 56, 28, 8, 1, 0, 1, 40, 360, 1546, 4144, 7896, 11408, 12866, 11440, 8008, 4368, 1820, 560, 120, 16, 1, 0, 1, 121, 2680, 27550, 180096, 866432, 3308736, 10453960, 27991600, 64472200, 129002640, 225783740, 347370800, 471435000, 565722640, 601080385, 565722720, 471435600, 347373600

Offset: 0

Paul D. Hanna, Jul 25 2009

From Manfred Boergens, Apr 07 2025: (Start)
T(n,k) is the number of collections of k [n]-subsets with union=[n]; with [0] = {}.
For n > 0: If more than half of the subsets are drawn their union covers [n] (see Formula). - The proof is based on 2^(n-1) being the number of subsets of [n] with one fixed element of [n] missing.
For collections of nonempty subsets see A055154.
For disjoint collections of subsets see A256894.
For disjoint collections of nonempty subsets see A008277. (End)

			Triangle begins:
  1,1;
  0,1,1;
  0,1,4,4,1;
  0,1,13,44,67,56,28,8,1;
  0,1,40,360,1546,4144,7896,11408,12866,11440,8008,4368,1820,560,120,16,1;
  ...

G. C. Greubel, Table of n, a(n) for the first 11 rows, flattened

Cf. A000371 (row sums), A381683 (partial row sums), A134174 (main diagonal).

Cf. A008277, A055154, A256894.

Table[Sum[(-1)^(n - j)*Binomial[n, j]*Binomial[2^j, k], {j, 0,
   n}], {n, 0, 5}, {k, 0, 2^n}]//Flatten (* G. C. Greubel, Dec 19 2016 *)

T(n,k)=sum(j=0,n,(-1)^(n-j)*binomial(n,j)*binomial(2^j,k))

T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(n,j)*C(2^j,k), k=0..2^n.
Row sums form A000371 (nondegenerate Boolean functions of n variables).
Main diagonal equals A134174 and is defined by the g.f.:
Sum_{n>=0} log(1 + (2^n-1)*x)^n/n!.
From Manfred Boergens, Apr 11 2024: (Start)
T(n,k) = A055154(n,k) + A055154(n,k-1) for n > 0, k > 0; A055154(n,j) are not defined for j = 0 and j = 2^n and are set = 0.
T(n,k) = C(2^n,k) for k > 2^(n-1).
T(n,k) < C(2^n,k) for k <= 2^(n-1), n > 0.
(Note: C(2^n,k) is the number of all k-subsets of P([n]).) (End)

A133990 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(2^k + n - 1,n).

Original entry on oeis.org

1, 1, 5, 71, 2747, 306861, 106709627, 123122238887, 492425723170553, 7012142056418141897, 361269845371107759765065, 68033187103968192731087467135, 47171609221094330538117045468744655

Offset: 0

Paul D. Hanna and Vladeta Jovovic, Jan 21 2008

			G.f.: A(x) = 1 + x + 5*x^2 + 71*x^3 + 2747*x^4 + 306861*x^5 +...
where
A(x) = 1 - log(1-x) + log(1-3*x)^2/2! - log(1-7*x)^3/3! + log(1-15*x)^4/4! - log(1-31*x)^5/5! + log(1-63*x)^6/6! - log(1-127*x)^7/7! + log(1-255*x)^8/8! +...

G. C. Greubel, Table of n, a(n) for n = 0..59

Cf. A134174.

A133990 := proc(n) add((-1)^(n-k)*binomial(n,k)*binomial(2^k+n-1,n),k=0..n) ; end: seq(A133990(n),n=0..15) ; # R. J. Mathar, Jan 30 2008

Table[Sum[(-1)^(n-k) Binomial[n,k]Binomial[2^k+n-1,n],{k,0,n}], {n,0,15}] (* Harvey P. Dale, Nov 24 2011 *)

{a(n)=local(A=1,X=x+x*O(x^n)); A=sum(k=0,n,log(1/(1-(2^k-1)*X))^k/k!); polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=1/n!*sum(k=0,n,(-1)^(n-k)*Stirling1(n,k)*(2^k-1)^n)}
for(n=0, 20, print1(a(n), ", "))

G.f.: Sum_{n>=0} (-log(1 - (2^n-1)*x))^n / n!.
a(n) = (1/n!) * Sum_{k=0..n} (-1)^(n-k) * Stirling1(n,k) * (2^k-1)^n.
From Vaclav Kotesovec, Jul 02 2016: (Start)
a(n) ~ binomial(2^n,n).
a(n) ~ 2^(n^2) / n!.
a(n) ~ 2^(n^2 - 1/2) * exp(n) / (sqrt(Pi) * n^(n+1/2)).
(End)

More terms from R. J. Mathar, Jan 30 2008

A133991 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(2^k+n-1,n).

Original entry on oeis.org

1, 3, 17, 193, 5427, 463023, 134675759, 139917028089, 527871326293913, 7281357469833220843, 368715613115281663650597, 68787958348542935934247206953, 47453320297069210448891035137347047

Offset: 0

Paul D. Hanna and Vladeta Jovovic, Jan 21 2008

Seiichi Manyama, Table of n, a(n) for n = 0..59

Cf. A133990, A134173, A134174.

Table[Sum[(-1)^(n-k) * StirlingS1[n, k]*(2^k + 1)^n, {k, 0, n}]/n!, {n, 0, 12}] (* Vaclav Kotesovec, Jun 08 2019 *)

a(n) = sum(k=0, n, binomial(n, k)*binomial(2^k+n-1, n)); \\ Seiichi Manyama, Feb 24 2023

a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n-k)*Stirling1(n,k)*(2^k+1)^n.
G.f.: Sum_{n>=0} (-log(1 - (2^n+1)*x))^n/n!.
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jun 08 2019

More terms from Alexis Olson (AlexisOlson(AT)gmail.com), Nov 14 2008

A136648 Inverse binomial transform of A014070: a(n) = Sum_{k=0..n} (-1)^(n-k)C(n,k)C(2^k,k).

Original entry on oeis.org

1, 1, 3, 43, 1625, 192785, 73792371, 94005141667, 408909577044065, 6204433373664395569, 334203804752658372354515, 64828498485572980097719939179, 45811084061472137471487315433296153, 119028111984311982345314987179033877373025, 1145664208319965667452046935744516601565935434531

Offset: 0

Paul D. Hanna and Vladeta Jovovic, Jan 21 2008

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A014070 (C(2^n, n)), A134174.

Table[Sum[(-1)^(n-k)*Binomial[n,k]*Binomial[2^k,k], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)

{a(n)=sum(k=0,n,(-1)^(n-k)*binomial(n,k)*binomial(2^k,k))}

/* Using the g.f.: */ {a(n)=my(X=x+x*O(x^n));polcoeff(sum(k=0,n,(log(1+(2^k+1)*X)-log(1+X))^k/k!)/(1+X),n)}

G.f.: A(x) = (1/(1+x))*Sum_{n>=0} [log(1 + (2^n+1)*x) - log(1+x)]^n / n!.

a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016

Edited by Charles R Greathouse IV, Oct 28 2009

Terms a(13) and beyond from Andrew Howroyd, Feb 02 2020

cp's OEIS Frontend

A163353 G.f.: A(x,y) = Sum_{n>=0,m>=0} (2^m-1)^n*x^n * log(1+y)^m/m!.

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A133990 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(2^k + n - 1,n).

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A133991 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(2^k+n-1,n).

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A136648 Inverse binomial transform of A014070: a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n,k)*C(2^k,k).

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A163353 G.f.: A(x,y) = Sum_{n>=0,m>=0} (2^m-1)^nx^n log(1+y)^m/m!.

A136648 Inverse binomial transform of A014070: a(n) = Sum_{k=0..n} (-1)^(n-k)C(n,k)C(2^k,k).